What it takes to be a giant

On a walk around the San Jose neighborhood, I encountered a single absolutely giant sunflower in a sidewalk garden.

giant sunflower

I admired the size of the flower head, which was about 16 inches across and probably weighed 10 pounds, wondering how many seeds must be packed in so very tightly and mathematically precisely (see an earlier post on “how many seeds in a sunflower seed head?”).

giant sunflower

Seeds are precisely arranged in spiral rows to maximize packing.

But then I got to thinking about what it takes to produce that giant flower head and develop all those seeds.  Supported by enhanced woody fibers in the stalk and fed by photosynthetic machinery in huge, oversized leaves and an elongated, deep taproot reaching deep into the soil for water and nutrients, the enormous reproductive output of this plant has the potential to be record-breaking.

But alas, a quick google search confirmed that Hans-Peter Schaffer holds the Guinness record for sunflower height (30 feet, 1 inch), mine was probably just over 8 feet. The giant Mongolian sunflowers routinely grow to 16-18 feet and sport 18-24 inch flower disks, so my giant wasn’t really record breaking at all.  Still impressive for an herbaceous plant, though!


Most animals seem to be attuned to recognizing another individual’s symmetry as a means of judging its fitness as a potential mate. Among most human cultures, both men and women will usually choose the most bilaterally (equal halves) symmetrical face as the most beautiful. Thus, it’s no surprise that we humans prize the beauty of a highly symmetrical flower as well:  e.g., orchids, with their perfect bilateral symmetry, or dahlias, with their fascinating radial symmetry.



While plant breeders may have selected for the perfect radial symmetry of a dahlia, which we love to show off in our gardens, other forces were at work in the production of the spiral pattern of the seeds in a sunflower head, or disc flowers of the Black-eyed Susan, Purple Coneflower, and Sunflowers.


Only one ring of yellow disc flowers are open for pollination at any one time.

Only one ring of yellow disc flowers are open for pollination at any one time.

purple coneflower

The arrangement of the emerging disc flowers is positioned so as to maximize their number in the available space.  When fertilized by pollinators, this will produce the maximum number of seeds per flower head — i.e., maximizing the fitness of that particular plant in its ability to pass on its genes.  To achieve this packing density, the disc flowers must lie at an angle of exactly 137.5 degrees from its neighbor, resulting in a series of spirals, rather than distinct horizontal or vertical rows. The same pattern is established in the growth of basal leaves of some plants (e.g., agave) and flower petals.

Chrysanthemum petals emerge in a spiral pattern, which reduces the overlap of petals in adjacent layers.

Chrysanthemum petals emerge in a spiral pattern, which reduces the overlap of petals in adjacent layers.

The spiral pattern is laid out with mathematical precision  — nature’s symmetrical design.  You can read more about the mathematical basis of the design in an earlier post.

Sunflower math

A golden rule of biology is that individuals attempt to leave as many offspring (i.e., genes) as possible in the next generation.  Plants, like the sunflowers growing in my backyard right now, try to maximize their seed production by packing seeds into the flower head in the most optimal way.  And that’s where the math comes in.

The sunflowers have grown quite tall (the fence posts are five feet), and some of the flowers are dinner-plate size (that’s a bumblebee on the flower head below for size reference).

But have you ever really inspected the interior of one of these complex flower heads?

The outer, yellow petals are really individual infertile ray flowers.  The center, disk flowers open from outer toward inner rows, a few layers at a time.  The yellow-tipped anthers stick up from the central part of each flower, presenting a disk of pollen for the lucky pollinators.

Once fertilized by the many bees that visit these flowers, each of the individual flowers produces one seed.  The entire flower head can produce as many seeds as there are individual disk flowers.  So the question is, how to pack as many of those flowers as possible into the circular head.

This pattern is not random.  Each disk flower (or potential seed) lies at an angle of 137.5 degrees from its neighbor.  Lined up from center to outer rim, the flowers (or seeds) describe two sets of spirals, one curving to the left, and one curving to the right.  This is the formula for maximal packing into circular space, and has been termed the “golden angle”.

As explained by mathematicians, divide a circle into two sections, such that the ratio of the large arc to the small arc is the same as the ratio of the entire circle to the large arc.  The golden angle is that created by the smaller of the two arcs, exactly 137.5 degrees.


Applying the golden angle to the construction of the flowerhead, we get the following sort of pattern.

Pretty darn smart of those plants!

A crop of sunflowers growing in northwestern Minnesota near Crookston.